Let \(K:=\ker \phi \) . Since the \(K\) -orbit of every \(x\in \mathbb {R}^N\setminus \lbrace 0\rbrace \)  is infinite, the embedding \(H^1(\mathbb {R}^N)^K\hookrightarrow L^p(\mathbb {R}^N)\)  is compact, where \(H^1(\mathbb {R}^N)^K\)  denotes the space of \(K\) -invariant functions in \(H^1(\mathbb {R}^N)\) ; see, e.g., [1]}. Since \(H^1(\mathbb {R}^N)^\phi \subset H^1(\mathbb {R}^N)^K\) , our claim follows.
